Fig 1.
a) RyR distribution in the cell. Each CaRU is formed by four simulation voxels, each one containing 9 RyRs. Thus, all CaRUs are formed by 36 RyRs. The CaRUs are distributed over the cell along the z-lines with a Gaussian distribution in both transversal and longitudinal axes. b) Each RyR follows a four state model, with stochastic transitions among the different states.
Fig 2.
Sketch of the different compartments considered in the simplified model, with the internal variables and the equations of the respective calcium fluxes.
Fig 3.
A: Calcium traces obtained with the full subcellular model and three different values of the average calcium concentration, . B: Line-scans at different values of the load. Increasing the load, the system undergoes a transition from a low cytosolic calcium state (at
), where RyRs remain in the closed state, to spontaneous oscillations, giving rise to calcium waves (
). Finally, at high calcium loads (
) oscillations give rise to a high cytosolic calcium state, where the RyRs remain open, resulting in SR calcium depletion.
Fig 4.
The average period of oscillations at different values of the average calcium concentration , for a concentration of CSQ of BSQ = 2mM (green dots), and in the absence of CSQ (blue dots).
Fig 5.
Time traces of the different calcium concentrations for different values of total calcium concentration, , and calsequestrin concentration set to zero (BSQ = 0).
After a transient, the system ends up in either a steady state which is excitatory at low levels of total calcium in the cell with observed low levels of calcium in the cytosol, in an oscillatory state with intermediate levels of total calcium in the cell, or in a state of high total levels of calcium in the cell with observed high cytosolic calcium levels.
Fig 6.
Plot of the function f(ci) for different values of the concentration.
a) At low concentrations there is a single fixed point. b) At higher concentrations two extra unstable fixed points appear. c) At high concentrations the upper fixed point becomes stable.
Fig 7.
Solutions for cytosolic calcium concentration, ci, as a function of total calcium concentration, .
Discontinuous lines represent unstable solutions while continuous lines stable ones.
Fig 8.
a) Solutions for cytosolic calcium concentration, ci, as a function of total calcium concentration, . Discontinuous lines represent unstable solutions and continuous lines stable ones. A closer look at the transitions is shown in b). When reducing the total concentration, at
, a limit cycle emerges in a Hopf bifurcation, from the upper state, that then becomes unstable. The red lines represent the lower and upper values of the limit cycle. At,
, the intermediate unstable fixed point collides with the limit cycle, that disappears in a homoclinic bifurcation. Below
, the RyR close state is the only solution. c) Oscillation periods as a function of
.
Fig 9.
a) Number of fixed points and stability of those fixed points as a function of total calcium in the unit, for different values of calsequestrin concentration. In b) and c) details of the transition are shown, for selected values of BSQ.
Fig 10.
Structure of the nullclines at different values of indicated in the title of each panel.
The black line indicates the first nullcline , while the orange lines corresponds to
. Dots indicate the fixed points. Filled dot: stable fixed point and unfilled dot: unstable fixed point.
Fig 11.
Structure of the nullclines at different values of indicated in the title of each panel.
The black line indicates the nullcline , while the orange line corresponds to
. The red curve is a trajectory with a direction indicated by the red arrows.
Fig 12.
Dependency of the onset of oscillations with buffer parameters.
The filled dots represent the control values Bb = 80μM, Kb = 0.5μM, given in Table 2 in S1 File.
Fig 13.
Structure of the nullclines at two different values of a) , b)
.
A black line corresponds to the nullcline , while the orange line indicates
. The functions f1 and g2 are the elements of the diagonal of the Jacobian matrix defined as
and
.
Fig 14.
Solutions as a function of total calcium concentration , with calsequestrin concentration set to zero when the fast dyadic approximation is used.
We obtain the same type of structure as expected. The system can be in a monostable state, which is excitatory (low load), in an oscillatory state (intermediate load), or in a bistable state (high loads), where it usually ends in a state of open RyR and depleted SR calcium concentration.
Fig 15.
Structure of the nullclines at different values of indicated in the title of each panel.
The black line indicates the nullcline , while the orange line corresponds to
. The red curve is a trajectory with a direction indicated with the red arrows.
Fig 16.
a) Traces of the cytosol and SR calcium concentrations for three different values of total calcium concentration . b) Oscillation periods as a function of
. The strength of the noise is σ = 2 ⋅ 10−3.